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A number of recent works have discussed the issue of spin polarization of aMajorana zero mode in condensed matter systems. Here we show that the spinpolarization density of a Majorana zero mode, computed as an average of thespin operator over its wave function, vanishes everywhere. A singlenon-degenerate Majorana zero mode, therefore, does not couple to an appliedmagnetic field, except via hybridization with higher energy excited states (ifpresent), which may perturb its wave function. If `spin' is defined byconsidering only the particle components of the wave function, as has been donein some recent works, Majorana zero modes do have a non-zero spatial profile ofthis quantity, measurable in scanning tunneling microscopy (STM) experiments.However, if such a quantity is measured in spin-resolved tunneling experiments(without spatial resolution), we show that it cannot be used as a uniquesignature of Majorana zero modes in the topologically non-trivialsuperconducting phase. As a byproduct, we show that in spatially inhomogeneoussystems, accidental zero energy modes, which for all practical purposes behaveas Majorana zero modes (including giving rise to a zero bias conductance peakof height 2e^2/h), can appear with increasing magnetic field even in theabsence of a topological quantum phase transition (TQPT). But only after gapclosing and the associated TQPT, the modes are localized near the system edges,resulting in the maximum topological protection. In the light of theseconsiderations, demonstrating the nonlocal character of thetopologically-protected Majorana pair and its emergence {\em after} the systemsundergo a TQPT, become critical tasks for the ongoing experimental search forMajorana bound states in condensed matter systems.

Spin polarization of Majorana zero modes and topological quantum phase transition in semiconductor Majorana nanowires